Chapter 8 Conservation of Energy Copyright © 2009 Pearson Education, Inc. Units of Chapter 8 • Conservative and Nonconservative Forces • Potential Energy • Mechanical Energy and Its Conservation • Problem Solving Using Conservation of Mechanical Energy • The Law of Conservation of Energy • Energy Conservation with Dissipative Forces: Solving Problems Copyright © 2009 Pearson Education, Inc. Units of Chapter 8 • Gravitational Potential Energy and Escape Velocity • Power • Potential Energy Diagrams; Stable and Unstable Equilibrium Copyright © 2009 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces A force is conservative if: the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken. Example: gravity. Copyright © 2009 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Another definition of a conservative force: a force is conservative if the net work done by the force on an object moving around any closed path is zero. Copyright © 2009 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force. Copyright © 2009 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Potential energy can only be defined for conservative forces. Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy An object can have potential energy by virtue of its surroundings. Familiar examples of potential energy: • A wound-up spring • A stretched elastic band • An object at some height above the ground Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy In raising a mass m to a height h, the work done by the external force is . We therefore define the gravitational potential energy at a height y above some reference point: . Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy This potential energy can become kinetic energy if the object is dropped. Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces). If Ugrav = mgy, where do we measure y from? It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured. Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy Example 8-1: Potential energy changes for a roller coaster. A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1. (b) What is the change in potential energy when the car goes from point 2 to point 3? (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3. Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy General definition of gravitational potential energy: For any conservative force: Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy A spring has potential energy, called elastic potential energy, when it is compressed. The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring. Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy Then the potential energy is: Copyright © 2009 Pearson Education, Inc. 8-2 Potential Energy In one dimension, We can invert this equation to find U(x) if we know F(x): In three dimensions: Copyright © 2009 Pearson Education, Inc. 8-3 Mechanical Energy and Its Conservation If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero—the kinetic and potential energy changes are equal but opposite in sign. This allows us to define the total mechanical energy: And its conservation: . Copyright © 2009 Pearson Education, Inc. 8-3 Mechanical Energy and Its Conservation The principle of conservation of mechanical energy: If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved. Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy In the image on the left, the total mechanical energy at any point is: Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-3: Falling rock. If the original height of the rock is y1 = h = 3.0 m, calculate the rock’s speed when it has fallen to 1.0 m above the ground. Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-4: Roller-coaster car speed using energy conservation. Assuming the height of the hill is 40 m, and the roller-coaster car starts from rest at the top, calculate (a) the speed of the roller- coaster car at the bottom of the hill, and (b) at what height it will have half this speed. Take y = 0 at the bottom of the hill. Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Conceptual Example 8-5: Speeds on two water slides. Two water slides at a pool are shaped differently, but start at the same height h. Two riders, Paul and Kathleen, start from rest at the same time on different slides. (a) Which rider, Paul or Kathleen, is traveling faster at the bottom? (b) Which rider makes it to the bottom first? Ignore friction and assume both slides have the same path length. Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Which to use for solving problems? Newton’s laws: best when forces are constant Work and energy: good when forces are constant; also may succeed when forces are not constant Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-6: Pole vault. Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter’s center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself. Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy For an elastic force, conservation of energy tells us: Example 8-7: Toy dart gun. A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (with spring stiffness constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length (x = 0), what speed does the dart acquire? Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-8: Two kinds of potential energy. A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring, which it compresses an amount Y = 15.0 cm. Determine the spring stiffness constant of the spring. Assume the spring has negligible mass, and ignore air resistance. Measure all distances from the point where the ball first touches the uncompressed spring (y = 0 at this point). Copyright © 2009 Pearson Education, Inc. 8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-9: A swinging pendulum. This simple pendulum consists of a small bob of mass m suspended by a massless cord of length l. The bob is released (without a push) at t = 0, where the cord makes an angle θ = θ0 to the vertical. (a) Describe the motion of the bob in terms of kinetic energy and potential energy. Then determine the speed of the bob (b) as a function of position θ as it swings back and forth, and (c) at the lowest point of the swing. (d) Find the tension in the cord, r FT. Ignore friction and air resistance. Copyright © 2009 Pearson Education, Inc. 8-5 The Law of Conservation of Energy Nonconservative, or dissipative, forces: Friction Heat Electrical energy Chemical energy and more do not conserve mechanical energy. However, when these forces are taken into account, the total energy is still conserved: Copyright © 2009 Pearson Education, Inc. 8-5 The Law of Conservation of Energy The law of conservation of energy is one of the most important principles in physics. The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one object to another, but the total amount remains constant. Copyright © 2009 Pearson Education, Inc. 8-6 Energy Conservation with Dissipative Forces: Solving Problems Problem Solving: 1. Draw a picture. 2. Determine the system for which energy will be conserved. 3. Figure out what you are looking for, and decide on the initial and final positions. 4. Choose a logical reference frame. 5. Apply conservation of energy. 6. Solve. Copyright © 2009 Pearson Education, Inc. 8-6 Energy Conservation with Dissipative Forces: Solving Problems Example 8-10: Friction on the roller-coaster car. The roller-coaster car shown reaches a vertical height of only 25 m on the second hill before coming to a momentary stop. It traveled a total distance of 400 m. Determine the thermal energy produced and estimate the average friction force (assume it is roughly constant) on the car, whose mass is 1000 kg. Copyright © 2009 Pearson Education, Inc. 8-6 Energy Conservation with Dissipative Forces: Solving Problems Example 8-11: Friction with a spring.